In communications networks, there may be a challenge to obtain good performance and capacity for a given communications protocol, its parameters and the physical environment in which the communications network is deployed.
For example, one parameter in providing good performance and capacity for a given communications protocol in a communications network is bandwidth. The bandwidth limits the information rate of a communication system and is a limited resource, hence it should be used efficiently. For this reason, faster-than-Nyquist (FTN) signaling has attracted interest as it enables a higher data rate (in comparison to schemes not using FTN signaling) without increasing the bandwidth, either by performing spectral compression or by transmitting pulses with a given time-domain shape at a rate that violates Nyquist's intersymbol interference (ISI) criterion. Since the pioneering work by Saltzberg (“Intersymbol interference error bounds with application to ideal bandlimited signaling,” IEEE Transactions on Information Theory, vol. 14, no. 4, pp. 563-568, 1968) and by Mazo (“Faster-than-Nyquist signaling,” Bell System Technical Journal, vol. 54, no. 8, pp. 1451-1462, 1975) researchers have been investigating the performance limits of FTN and methods for practical implementation.
FTN has traditionally required the use of the Viterbi algorithm for reliable decoding, an approach whose complexity grows exponentially with the number of interfering symbols and polynomially with the number of points in the signal constellation. As an example, decoding a M-PAM (where PAM is short for pulse-amplitude modulation) signal with N symbols and an effect of ISI lasting K symbols results in a computational complexity of O(N·MK). For large K and/or M this algorithm becomes impractical to be used.
Using the Viterbi algorithm for data decisions in the presence of ISI is thus computationally demanding if the modulation order is high (i.e., the value of M is large) and/or there is ISI interaction between many symbols (i.e., the value of K is large).
An FTN scheme has been proposed which is not based on the Viterbi algorithm and instead uses precoding in the transmitter and a postcoding in the receiver to allow for optimal data decisions in the presence of ISI. This FTN scheme operates on a block of N data symbols and uses the inverse square root of the Gram matrix of the communication channel to precode the symbol amplitudes in the transmitter and perform postcoding in the receiver after matched filtering. The computational complexity of the pre- and postcoding operations is O(N2) regardless of the number of constellation points (i.e., regardless of M) and calculating the pre/postcoding matrix has a computational complexity of O(N3). This FTN scheme requires storing the N×N coding matrix on which the Gram matrix is based in both the transmitter and the receiver (although the actual number of elements to be stored is closer to N2/2 due to symmetry properties of the N×N coding matrix).
However, the above disclosed PIN scheme based on precoding and postcoding requires a computationally demanding inversion of the square root of the channel Gram matrix which has O(N3) complexity and can be difficult to perform on the fly if the FTN parameters has to be changed. This could be the case if, e.g., the channel conditions change to become more/less demanding. Furthermore, according to the above disclosed FTN scheme the precoding is applied as a matrix multiplication on the complete block of data to be transmitted. This makes the above disclosed FIN scheme unfeasible for transmitting long data sequences, such as data sequences containing thousands or more information bits. Further, the large matrix would consume a lot of memory in both the transmitter and the receiver.
Hence, there is still a need for an improved precoding (and postcoding) for faster-than-Nyquist communications.